Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 2 : Chapter 5

Lecture 3

IMPORTANT FORMULAS
Scaled Scores:

The raw score is an untransformed score from a measurement.

Raw Data

Statistical data in its original form, before any statistical techniques are used to refine, process, or summarize.

Is 85/100 correct on an exam good? It depends on the evaluation system.

Raw scores are difficult to interpret without additional information.
"The curve" has preset percentage categories for evaluation.
For Ex. A score in the top 10% on an exam = an 'A'. It is a relative position to other test scores.
"The standard" has preset cutoffs (relative to cut off values).
For Ex. 100-90 points correct on an exam = an 'A'. With the standard you are up against a system, not each other.

Percentile

One of the division points between 100 equal-sized pieces of the population when the population is arranged in numerical order. The 78th percentile is the number such that 78% of the population is smaller and 22% of the population is larger.

The percentile is transformed from a raw score. It will give you a relative position, for example, 1 to 99. The numbers = the percentage of scores below your raw score.

Obtaining a percentile rank of 80 means that whatever your raw score was, 80% of the other raw scores were below yours.

Raw Score	Percentile
78	50
85	60
90	70

Both sides (the distance between the scores) need to be equal for a linear transformation. The big problem with percentiles is that they are not linear transformations from raw scores, hence they are called "non-linear".

Centimeter	Inch
2.54	1
5.08	2

These are linear transformations.
As one side changes the other changes in equal proportions.
Percentiles are great at a descriptive level.

Standard Scores are linear transformations.

Raw scores are transformed into standard scores.
Changing a raw score to a percentile score or percentile rank:


Given a set of scores: 52, 89, 42, 13, 88, 76, 44, 45, 22, 105
Find the percentile rank (PR) for 45.

Lecture 4

Scaled Scores:
Percentile rank
Standard score (a linear transformation)

Z= raw score - mean of raw scores/standard deviation of raw scores

The Z score tells you how far the raw score is away from the mean in terms of standard deviation units.
It does not change the shape of the distribution!

Raw score does not change into a bell shaped curve when changed into standard scores.
The numbers do not change physically, the measurement just changes.
It transforms the unit of measurement you are working with.

Percentile
Z score
mean of Z scores = 0
standard deviation of Z scores = 1
If the Z score is negative, this says that the raw score was below the mean of raw scores.
If the Z score is positive, this says that the raw score was above the mean of raw scores.
If the Z score is zero, this says that the raw score was equal to the mean of raw scores.

"CEEB Score"

The College Examination Board (now called ETS) converts your raw score into a scale (different unit of measurement) where the mean is 500 and the standard deviation is 100. This is done to eliminate negative numbers.

SAT	Lowest	Highest	Mean
Verbal	200	800	500
Quantatative	200	800	500

The Graduate Entrance Examination (GRE) is the same.

McCall T-Score Scale (Personality Tests)

The raw scores have been converted resulting in a scale where the mean = 50 and the standard deviation = 10.

Ex. Given the raw scores 22, 17, 19, 37, 26 convert 26 to a scaled score on the McCall T-Score Scale.

Ex. -23, 18, -2, 5, 44, 39, 19, 18 . What is the percentile rank for a raw score of 18 (X= 18).

Tied Ranks are 2 numbers having the same values, for example '18' and '18'.

Rank	Scores
1	-23
2	-2
3	5
4	18
5	18
6	19
7	39
8	44

Look at ranks 4 and 5, take the mean of these = 4.5.

Ex. 4, 29, 17, 29, 29, 30, 7, 11, 14. Convert 29 to a percentile.

Rank	Scores
1	4
2	7
3	11
4	14
5	17
6	29
7	29
8	29
9	30

Dr. Lee may use different wording on your exams. Don't get confused. Percentile and Percentile Rank mean the same thing.

Lecture 5

X (raw scores)	Rank
100	7
80	6
29	2
46	5
18	1
45	4
34	3

1) Rank the numbers.

What does this mean?
It means that 64% of the scores lie below 46.

Ex. Convert 80 to a Z score.

M = Mean
X-bar is also used but not in this class because it can be confused with X which is a vector.

Convert 80 to a Z score.

1. Find the mean and standard deviation.

M = 50.285
SD = 27.154

Z = 80 - 50.28 /27.154

Z = 1.094 = 1.09
This says that the score of 80 lies over 1 standard deviation above the mean (50.285).

The following is very important:
Percentiles are represented as integers.
Z scores are carried to 2 decimal places.
To insure accuracy at 2 decimal places you must carry to at least 3 decimal places in your calculations.

Ex. Convert 46 to a Z score.

Z = 46 - 50.285 / 27.154

46 is .16 standard deviations below the mean.

Ex. Convert all the scores to Z scores.
What is the mean of the Z scores?
The mean = 0.

* The mean of the Z scores is zero.
* The standard deviation of the scores is 1.

Ex. Convert 80 to a CEEB scaled score.

Use this formula :
CEEB = 100 ( Z score for 80) + 500

1. Find the Z score for 80.
2. use the formula listed above.
100 (1.09) + 500 = 609.

Ex. Convert 80 to a McCall T- score scale.
T score = 10 ( Z score for 80) + 50
10 (1.09) + 50 = 10.9 + 50 = 60.9.

Ex. Convert X = 80 to a scaled score where the mean = 100 and standard deviation = 16.
Scaled score = 16 ( Z score for 80) + 100
Scaled score = 16 ( 1.09) + 100 = 117.44.